Journal ArticleJournal of Algebra · April 15, 2019
For a scheme X defined over the length 2 p-typical Witt vectors W2(k) of a characteristic p field, we introduce total p-differentials which interpolate between Frobenius-twisted differentials and Buium's p-differentials. They form a sheaf over t ...
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Journal ArticleMathematische Annalen · December 1, 2018
We define a tropicalization procedure for theta functions on abelian varieties over a non-Archimedean field. We show that the tropicalization of a non-Archimedean theta function is a tropical theta function, and that the tropicalization of a non-Archimedea ...
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ConferenceProceedings of Symposia in Pure Mathematics · January 1, 2018
We describe recent work connecting combinatorics and tropical/ non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of Chabauty–Coleman lies at ...
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Journal ArticleAnnales De L Institut Fourier · January 1, 2017
In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let K be a complete non-Archimedean field, and let X be a closed subscheme of a toric variety over K. We define the tropical skeleton of ...
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Journal ArticleCanadian Journal of Mathematics · December 1, 2016
Let J be a Jacobian variety with toric reduction over a local field K. Let J → E be an optimal quotient defined over K, where E is an elliptic curve. We give examples in which the functorially induced map φJ → φE on component groups of the Neron ...
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Journal ArticleAdvances in Mathematics · May 14, 2016
Let K be a complete, algebraically closed non-archimedean field with ring of integers K∙ and let X be a K-variety. We associate to the data of a strictly semistable K∙-model X of X plus a suitable horizontal divisor H a skeleton S(X,H ...
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Journal ArticleAlgebraic Geometry · January 1, 2016
We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev multiplicity form ...
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Journal ArticleDuke Mathematical Journal · January 1, 2016
Let X be a curve of genus g ≥ 2 over a number field F of degree d = [F : Q]. The conjectural existence of a uniform bound N (g, d) on the number #X(F) of F-rational points of X is an outstanding open problem in arithmetic geometry, known by the work of Cap ...
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Journal ArticleResearch in Mathematical Sciences · December 1, 2015
Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certain combinatorial objects, called skeleta. A skeleton is a me ...
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Journal ArticleInternational Mathematics Research Notices · January 1, 2015
Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let be its value group. Given a smooth, proper, connected K-curve X and a skeleton of the Berkovich analytification Xan, there are two natu ...
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Journal ArticleAlgebra and Number Theory · January 1, 2015
We prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the nonvanishing of certain Hurwitz numbers, and we give ...
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Journal ArticleJournal of the Institute of Mathematics of Jussieu · April 1, 2012
Let R be a complete rank-1 valuation ring of mixed characteristic (0, p), and let K be its field of fractions. A g-dimensional truncated Barsotti-Tate group G of level n over R is said to have a level-n canonical subgroup if there is a K-subgroup of G ⊗ - ...
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Journal ArticleAdvances in Mathematics · April 1, 2012
In this paper we use the connections between tropical algebraic geometry and rigid-analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: i ...
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Journal ArticleDiscrete Applied Mathematics · May 15, 2004
In their paper (Inform. Process. Lett. 77 (2001) 261), Wongngamnit and Angluin introduced a memory-efficient robot, called the Homing Robot, which localizes in an occupancy grid. We present a more general class of grids called hybrid grids, and establish t ...
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